Pathwise regularisation by noise in PDEs

Chapters

Appears in collection : Averaging and homogenization in deterministic and stochastic systems / Moyennisation et homogénéisation dans les systèmes déterministes et stochastiques

We discuss some examples of the "good" effects of "very bad", "irregular" functions. In particular we will look at non-linear differential (partial or ordinary) equations perturbed by noise. By defining a suitable notion of "irregular" noise we are able to show, in a quantitative way, that the more the noise is irregular the more the properties of the equation are better. Some examples includes: ODE perturbed by additive noise, linear stochastic transport equations and non-linear modulated dispersive PDEs. It is possible to show that the sample paths of Brownian motion or fractional Brownian motion and related processes have almost surely this kind of irregularity. (joint work with R. Catellier and K. Chouk)

Information about the video

Date of recording 13/05/2015
Date of publication 28/05/2015
Institution CIRM
Licence CC BY NC ND
Language English
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.18762503
Cite this video Gubinelli, Massimiliano (13/05/2015). Pathwise regularisation by noise in PDEs. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18762503
URL https://dx.doi.org/10.24350/CIRM.V.18762503

Domain(s)

Bibliography

Beck, L., Flandoli, F., Gubinelli, M., & Maurelli, M. (2014). Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness. <arXiv:1401.1530> - http://arxiv.org/abs/1401.1530
Catellier, R. (2015). Rough linear transport equation with an irregular drift. <arXiv:1501.03000> - http://arxiv.org/abs/1501.03000
Catellier, R., & Gubinelli, M. (2014). Averaging along irregular curves and regularisation of ODEs. <arXiv:1205.1735> - http://arxiv.org/abs/1205.1735
Chouk, K., & Gubinelli, M. (2015). Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations. <arXiv:1303.0822> - http://arxiv.org/abs/1303.0822
Chouk, K., & Gubinelli, M. (2014). Nonlinear PDEs with modulated dispersion II: Korteweg--de Vries equation. <arXiv:1406.7675> - http://arxiv.org/abs/1406.7675